Flag manifolds and the Landweber–Novikov algebra

Abstract

We investigate geometrical interpretations of various structure maps associated with the Landweber–Novikov algebra $S^{\ast }$ and its integral dual $S_{\ast }$. In particular, we study the coproduct and antipode in $S_{\ast }$, together with the left and right actions of $S^{\ast }$ on $S_{\ast }$ which underly the construction of the quantum (or Drinfeld) double $\mathcal{D}\left(S^{\ast }\right)$. We set our realizations in the context of double complex cobordism, utilizing certain manifolds of bounded flags which generalize complex projective space and may be canonically expressed as toric varieties. We discuss their cell structure by analogy with the classical Schubert decomposition, and detail the implications for Poincaré duality with respect to double cobordism theory; these lead directly to our main results for the Landweber–Novikov algebra.